Deep Koopman Approach for Nonlinear Dynamics

• Deep Koopman approach is a data-driven method that learns nonlinear lifts to a latent space where dynamics evolve linearly.
• It combines deep neural lifting with traditional Koopman operator theory to enable multi-step control and prediction.
• The approach effectively integrates noise handling, uncertainty quantification, and online adaptation for robust system modeling.

The deep Koopman approach denotes a family of data-driven methods that combine Koopman operator theory with deep neural networks to construct finite-dimensional lifted representations in which nonlinear dynamics evolve approximately or exactly linearly. Across the literature, the central object is a learned observable or latent state—variously written as $z_k$, $x_k$, $y_k$, $g(x_k)$, or $\Phi(x_k)$—for which the dynamics admit a predictor of the form $z_{k+1}=Az_k+Bu_k$, or controlled and stochastic generalizations thereof, while the nonlinear lifting itself is learned from data rather than specified by a hand-crafted dictionary (Morton et al., 2019, Shi et al., 2022, Iacob et al., 13 Jul 2025). In this sense, the approach sits between classical EDMD-style Koopman approximation, which fixes observables a priori, and unconstrained black-box sequence modeling, which does not impose latent linear structure (Han et al., 2020, Valábek et al., 6 Nov 2025).

1. Conceptual basis

The common theoretical premise is that a nonlinear dynamical system can be represented linearly in a lifted space of observables. For autonomous dynamics $x_{k+1}=f(x_k)$, the Koopman operator acts on observables $\phi$ by composition, $\mathcal K\phi=\phi\circ f$, and with a finite dictionary $$\Phi(x)=\big[\phi_1(x),\dots,\phi_{n_f}(x)\big]^\top$$, one seeks a lifted state $x_k$0 satisfying $x_k$1 under exact finite invariance (Iacob et al., 13 Jul 2025). For controlled systems, the finite-dimensional approximation is often written as

$x_k$2

or, in notation used in several papers,

$x_k$3

with the lifting $x_k$4 or $x_k$5 learned from data rather than chosen manually (Han et al., 2020, Hao et al., 2024).

A central theme in the deep literature is that the observable map is no longer fixed. Instead, neural networks parameterize the lifting, and sometimes the inverse lifting, so that the latent coordinates are optimized jointly with the latent linear dynamics. In some works this appears as an encoder–linear-dynamics–decoder architecture, while in others the original state is concatenated directly into the lifted state, so that reconstruction is given by a fixed linear projection $x_k$6 (Shi et al., 2022, Abtahi et al., 4 Mar 2025, Wang et al., 13 Jun 2026). This suggests a useful distinction between autoencoder-style deep Koopman models, which learn both lifting and reconstruction, and state-preserving lifted models, which enforce direct recoverability of the physical state.

The literature also emphasizes that finite-dimensional Koopman models are approximations except for special nonlinear classes. One paper explicitly notes that the lifted linear model is behaviorally equivalent to the original nonlinear system only on a nonlinear constraint manifold in lifted space (Iacob et al., 13 Jul 2025). A plausible implication is that deep Koopman models should be interpreted not as exact global linearizations in general, but as structured approximations whose value depends on how well the learned observables align with invariant or nearly invariant subspaces.

2. Representation learning architectures

A standard deep Koopman architecture learns a nonlinear map from measured states or outputs into a latent space where evolution is linear. In the Deep Variational Koopman model, the learned quantities are distributions over latent observations $x_k$7, from which linear latent dynamics matrices are computed by pseudoinverse regression, yielding a distribution over linear models rather than a single deterministic one (Morton et al., 2019). In the Deep Koopman Network for soft robot dynamics, the encoder maps state and control into latent coordinates intended to behave like Koopman eigenfunctions, the latent operator is parameterized through complex conjugate eigenvalue pairs, and the decoder reconstructs the physical state (Komeno et al., 2022). In other lines of work, the encoder is a multilayer perceptron $x_k$8, the latent dynamics are linear, and the decoder is either linear or nonlinear depending on whether prediction or control is the primary objective (Valábek et al., 6 Nov 2025, Garmaev et al., 2023).

Several formulations preserve the original state explicitly inside the lifted state. One controlled deep Koopman model defines

$x_k$9

so that the learned latent coordinates extend the state rather than replacing it (Shi et al., 2022). The MDK-Net vehicle model uses the same structural idea, writing $y_k$0 with $y_k$1, and then predicting with $y_k$2 (Abtahi et al., 4 Mar 2025). The adaptive vehicle model in 2026 similarly writes $y_k$3 and recovers the physical state by $y_k$4 with $y_k$5 (Wang et al., 13 Jun 2026). These designs are explicitly control-oriented because quadratic costs and state constraints can be expressed directly in physical coordinates embedded in the latent state.

Not all deep Koopman approaches learn the latent evolution in the same way. Some enforce linearity by construction through periodic coordinates. Deep Probabilistic Koopman uses

$y_k$6

and lets a neural network map these Koopman-style periodic coordinates to the parameters of a predictive distribution, thereby producing long-horizon probabilistic forecasts without iterative autoregressive rollout (Mallen et al., 2021). In contrast, hybrid EDMD–deep reinforcement learning work retains a hand-crafted polynomial dictionary for the Koopman part and uses a deep actor–critic network only to learn a residual correction term in the reduced lifted dynamics (Sun et al., 2024). This suggests that the phrase “deep Koopman approach” covers both fully neural observable learning and hybrid architectures in which deep networks augment, rather than replace, classical Koopman approximation.

3. Controlled and control-oriented formulations

Control is incorporated in several distinct ways. The most common is the lifted linear controlled system

$y_k$7

used for controller synthesis by LQR or MPC after learning a lifting map from data (Han et al., 2020, Valábek et al., 6 Nov 2025). In the Deep Koopman Representation for Control framework, the neural network learns observables $y_k$8, the lifted dynamics are $y_k$9 around a shifted operating point, and standard linear controllers such as MPC and LQR are applied in the lifted space (Han et al., 2020). In process control, the pasteurization-unit EMPC work trains a deep Koopman model with linear latent dynamics and a linear output map $g(x_k)$0, specifically so that the online controller remains a convex quadratic program (Valábek et al., 6 Nov 2025).

A second family addresses systems where the control channel is itself nonlinear or state-dependent. One such model generalizes the standard controlled Koopman form by learning a transformed control variable: $g(x_k)$1 and instantiates this in three variants: a standard linear-input model, a state-dependent affine-input model $g(x_k)$2, and a fully nonlinear control lifting $g(x_k)$3 (Shi et al., 2022). The DKAC and DKN variants were introduced specifically to address systems with terms such as $g(x_k)$4, where a strict $g(x_k)$5 channel is too restrictive (Shi et al., 2022). A related but distinct strategy appears in degradation modeling, where the Koopman-Inspired Degradation Model uses control-dependent latent operators $g(x_k)$6 and a control-conditioned decoder to separate degradation progression from direct control effects on observed outputs (Garmaev et al., 2023).

Some works keep the latent dynamics linear in the autonomous component but make the input channel more expressive. The noise-aware identification framework for nonlinear systems with inputs writes

$g(x_k)$7

and discusses linear, bilinear, input-affine, and fully nonlinear choices for $g(x_k)$8 and $g(x_k)$9 as a model-structure selection problem (Iacob et al., 13 Jul 2025). The Bouc–Wen hysteresis benchmark in that work showed that fully LTI input structure performs poorly, bilinear structure improves with higher lifting dimension but saturates, and input-affine or more general structures give the best trade-offs (Iacob et al., 13 Jul 2025). A plausible implication is that, in deep Koopman control-oriented identification, the expressive burden often shifts from the autonomous latent dynamics to the way inputs are parameterized in the lifted space.

Another control-related distinction concerns whether control is modeled as an external affine input or as part of an augmented autonomous system. The soft robotics spectral-analysis work explicitly treats control as one of the lifted variables, learning $\Phi(x_k)$0 and then applying sampling-based MPC with the learned model rather than LQR in a fixed $\Phi(x_k)$1 latent system (Komeno et al., 2022). This sacrifices the standard affine control form but preserves a spectral interpretation of the latent coordinates.

4. Noise, uncertainty, and online adaptation

A major recent development is the extension of deep Koopman methods beyond noiseless, fully observed trajectory fitting. One line of work studies bounded measurement noise. DKND starts from the baseline model

$\Phi(x_k)$2

and modifies the training objective by adding a robustness term derived from the discrepancy between clean and noisy Koopman representations under the assumption $\Phi(x_k)$3 (Hao et al., 2024). Its main contribution is therefore not a new latent architecture, but a noise-aware regularization of observable learning. The reported experiments show that DKND consistently outperforms standard deep Koopman learning on most noisy benchmarks and strongly outperforms DMDTLS in those settings (Hao et al., 2024).

A stronger statistical treatment appears in the identification framework for general noise conditions. That method models nonlinear systems with inputs, partial observations, process noise, and measurement noise through an innovation-form lifted predictor,

$\Phi(x_k)$4

and uses a reconstructability-based deep encoder to infer the latent state from finite histories of inputs and outputs (Iacob et al., 13 Jul 2025). Training is based on a squared multi-step prediction error in a multiple-shooting formulation, which reduces gradient-path length, enables parallel computation over subsequences, and yields statistical consistency under assumptions including existence of an exact finite-dimensional Koopman embedding, stability, quasi-stationary input, and weak persistent excitation (Iacob et al., 13 Jul 2025). This paper is unusual in the deep Koopman literature because consistency is stated with respect to an equivalence class of predictors rather than a unique parameter vector.

A separate literature treats uncertainty probabilistically rather than robustly. The Deep Variational Koopman model infers distributions over latent observations, which induces a distribution over latent linear systems $\Phi(x_k)$5 and therefore a distribution over future trajectories (Morton et al., 2019). Deep Probabilistic Koopman instead keeps the latent evolution explicitly periodic and uses neural networks to output time-varying parameters of a predictive distribution, thereby producing calibrated long-range probabilistic forecasts and uncertainty estimates (Mallen et al., 2021). These two works illustrate two different probabilistic interpretations of the deep Koopman idea: uncertainty over latent linear models versus uncertainty in output distributions generated from fixed linear latent evolution.

Noise also interacts with operator bias. The Deep Robust Koopman Network argues that nonlinear lifting transforms zero-mean state noise into lifted-space noise with nonzero mean, so least-squares estimation of the Koopman operator becomes systematically biased (Singh et al., 5 Jan 2026). It therefore learns forward and backward latent dynamics,

$\Phi(x_k)$6

and synthesizes a reduced-bias operator via

$\Phi(x_k)$7

Under first-order perturbation analysis, the paper states that this reduces the bias in expectation by roughly a factor of two (Singh et al., 5 Jan 2026). A plausible implication is that, for deep Koopman models trained on noisy robotics data, the operator-estimation problem may be at least as important as the representation-learning problem.

Online and adaptive variants have recently become more prominent. Distributed Deep Koopman Learning extends the framework to networks of agents with partial observations, where each agent learns a local encoder $\Phi(x_k)$8 but consensus is enforced on the shared lifted dynamics matrices $\Phi(x_k)$9 through distributed optimization and consensus iterations (Hao et al., 2024). COLoKe adds a conformal-style online update trigger so that Koopman embeddings are refined only when the current model’s prediction error exceeds a dynamically calibrated threshold (Gao et al., 16 Nov 2025). The adaptive vehicle model of 2026 identifies a different online bottleneck: the high-dimensional lifted space induces a rank-deficient update problem for recursive least-squares-type adaptation, and it replaces inverse-based updates with a Physics-Informed Variable Step-Size Normalized Least Mean Squares rule

$z_{k+1}=Az_k+Bu_k$0

where the anchor term suppresses drift toward unphysical operators (Wang et al., 13 Jun 2026).

5. Training objectives and empirical domains

Deep Koopman models are trained with a wide range of objectives, but several recurring patterns appear. Multi-step consistency is especially prominent. The MDK-Net vehicle model combines a single-step lifted loss,

$z_{k+1}=Az_k+Bu_k$1

with a multi-step rollout loss and a stability penalty on eigenvalues of $z_{k+1}=Az_k+Bu_k$2 outside the unit disk (Abtahi et al., 4 Mar 2025). The controlled deep Koopman paper of 2022 similarly trains with a $z_{k+1}=Az_k+Bu_k$3-step latent prediction loss, arguing that one-step fitting is insufficient for long-horizon prediction and control (Shi et al., 2022). In degradation modeling, the original DKO objective is explicitly

$z_{k+1}=Az_k+Bu_k$4

with $z_{k+1}=Az_k+Bu_k$5 and $z_{k+1}=Az_k+Bu_k$6 (Garmaev et al., 2023). These formulations all train the latent model for rollout consistency rather than only for one-step regression.

Empirical applications span a broad range of domains. In soft robotics, deep Koopman models were used both for spectral interpretation of oscillatory modes and for sampling-based MPC stabilization of a soft inverted pendulum (Komeno et al., 2022). In process systems, a deep Koopman EMPC for a laboratory pasteurization unit achieved a 45% improvement in open-loop prediction accuracy over N4SID for the linear-projection control model, a 32% reduction in total economic cost, and 10.2% less electrical energy in steady-state operation (Valábek et al., 6 Nov 2025). In degradation modeling, DKO and KIDM were applied to CNC milling cutters and Li-ion batteries, where KIDM+LR substantially outperformed AE+LR and KIDMAE+LR under varying load and gave an order-of-magnitude lower MSE in an early-life RUL experiment (Garmaev et al., 2023). In time-series forecasting, Deep Probabilistic Koopman outperformed all 177 competitors in the cited Global Energy Forecasting Competition setting, with a mean relative improvement of 15.4% over the vanilla benchmark while using only the demand series itself (Mallen et al., 2021).

Vehicle dynamics has become a particularly active application area. MDK-Net learns a lifted linear predictor in Frenet coordinates from low-level actuation inputs and uses it in MPC, reporting much lower 80-step open-loop MSE than an identified LTI model in a double lane-change scenario (Abtahi et al., 4 Mar 2025). The adaptive vehicle model of 2026 adds tire-force-informed offline training and stable online adaptation, reporting robust prediction accuracy under unseen excitations and an average HIL execution time of 0.421 ms (Wang et al., 13 Jun 2026). These results suggest that deep Koopman methods have become especially attractive where one wants the nonlinear expressivity of learned representations together with the optimization structure of linear prediction models.

6. Limitations, controversies, and open directions

The literature is explicit that deep Koopman methods do not remove the fundamental difficulty of finding finite-dimensional invariant subspaces. Several papers note that exact finite-dimensional Koopman embeddings are guaranteed only for special nonlinear classes, and that most learned models are therefore approximate (Iacob et al., 13 Jul 2025, Han et al., 2020). This has two consequences. First, prediction quality depends strongly on lifting dimension, horizon length, lag structure, and the parameterization of the input and noise channels. Second, linear latent structure may improve extrapolation relative to black-box models without guaranteeing global validity.

A second recurring limitation is that many papers remain strong on empirical performance and weaker on guarantees. The DKRC control paper provides a controllability-oriented penalty but does not analyze observability formally and does not offer guarantees on stability, robustness, or recursive feasibility (Han et al., 2020). The pasteurization EMPC paper gives a complete controller formulation but no explicit training loss, optimizer hyperparameters, solver-time statistics, or formal proof of closed-loop stability (Valábek et al., 6 Nov 2025). The soft-robot spectral-analysis paper offers compelling modal interpretations, but the link between latent modes and independent physical mechanisms remains qualitative (Komeno et al., 2022). This suggests that the deep Koopman literature is still split between representation-centric empirical practice and system-identification-style theory.

Noise and partial observation remain controversial points. Classical DMD/EDMD-style procedures often rely on directly measured states or fixed observable dictionaries and may produce biased predictors under process and measurement noise (Iacob et al., 13 Jul 2025). Many deep autoencoder Koopman models are deterministic and can confound latent dynamics with noise under partial observations (Iacob et al., 13 Jul 2025). Recent work addresses this through bounded-noise regularization, innovation-form predictors, forward–backward bias reduction, or probabilistic latent models, but these solutions introduce their own assumptions: bounded noise, temporal invertibility, exact model-class inclusion, full-row-rank lifted data matrices, or tractable innovation structure (Hao et al., 2024, Singh et al., 5 Jan 2026, Iacob et al., 13 Jul 2025).

A final open direction concerns online adaptability without catastrophic drift. Streaming settings introduce overfitting, selective update, and numerical stability issues that do not appear in offline training. COLoKe addresses this with conformal-style update triggering (Gao et al., 16 Nov 2025); distributed deep Koopman learning addresses it through consensus over shared lifted dynamics under partial observability (Hao et al., 2024); adaptive vehicle modeling addresses it through minimum-norm online updates in rank-deficient lifted spaces (Wang et al., 13 Jun 2026). A plausible implication is that the next phase of the deep Koopman approach will be defined less by discovering ever more expressive latent spaces and more by maintaining those embeddings safely under nonstationarity, partial observation, and streaming data.

Taken together, the deep Koopman approach has evolved from neural lifting for finite-dimensional linear latent dynamics into a broader research program spanning control, system identification, uncertainty quantification, degradation modeling, probabilistic forecasting, distributed learning, and adaptive online estimation. What unifies these variants is not a single architecture, but a shared methodological commitment: learn observables from data so that nonlinear dynamics can be treated through the lens of linear evolution in a lifted space (Morton et al., 2019, Shi et al., 2022, Iacob et al., 13 Jul 2025).